A Parallel High-Order Solver for Linear Elasticity Problems Using A Weak Galerkin Finite Element Method on Unstructured Quadrilateral Meshes by Yunze Li B.S. in Mechanical Engineering, July 2016, Xi’an Jiaotong University A Thesis submitted to The Faculty of The School of Engineering and Applied Science of The George Washington University in partial fulfillment of the requirements for the degree of Master of Science May 19, 2019 Thesis directed by Chunlei Liang Associate Professor of Mechanical and Aerospace Engineering c Copyright 2019 by Yunze Li All rights reserved ii Acknowledgments First and foremost, I want to extend my heartfelt gratitude to my supervisor, Dr Chunlei Liang, whose patient guidance and valuable suggestions help me significantly to complete this thesis. He has been a great example inspiring me to be a future scholar in the field of scientific computation. Moreover, I would like to thank Dr Lin Mu. She gave me tremendous encouragement and technical instruction in learning weak Galerkin finite element method. I am also greatly indebted to my lab mates and best friends: Mao Li, Xiaoliang Zhang and Bin Zhang, who have shared with me a lot of great time together in the past two years. Last but not least, I am deeply indebted to my parents, who always support me without any condition. iii Abstract A Parallel High-Order Solver for Linear Elasticity Problems Using A Weak Galerkin Finite Element Method on Unstructured Quadrilateral Meshes Weak Galerkin Finite Element Method (WGFEM) [15] is a high-order discontinuous method for unstructured grids. Recently, The WGFEM has been successfully applied to solve lineanr elasticity problems. In this work, we integrate a domain decomposition method with a WGFEM solver for parallel solutions of linear elasticity equations on unstructured quadrilateral grids. In particular, we employ a Balancing Domain Decomposition by Con- straints (BDDC) [4] to effectively reduce the computational cost of the coarse problem for the interface unknowns. The WG-BDDC method shares some similarity with the well- known FETI-DP. However, the standard continuous Galerkin Finite Element Method solves unknowns on nodal points. These nodal points on subdomain interfaces may connect to multiple elements in the FETI-DP method after domain decomposition. The WG-BDDC method does not need to involve solutions on the nodal points. Therefore, the communi- cation between two adjacent subdomains only requires to collect information over their common faces. A unique feature of the WGFEM method is the use of weak gradient operator to differentiate basis functions. For 2D problems, an integration by parts was used to change surface inte- grals to line integrals. WGFEM employs two kinds of basis functions. The first group is on the edges of each element and the second group is inside each element. Two groups of bsas function can be chosen independently. In order to have continuous solution in each element given that two different spaces are used, a stabilizer is introduced at element interfaces. In the BDDC method, the unknowns are grouped to interior, dual and primal spaces for each subdomain by the technique of Schur complement. Unknowns in the primal spaces will constitute a new global problem to solve. The size of this new global problem is significantly smaller than that of the global problem involving all unknowns on subdomain interfaces. First, we tested the orders of accuracy of WG-BDDC method on structured grids. Subse- iv quently, by using our university cluster, the WG-BDDC is robust for solving linear elasticity problems on parallel computers. Excellent scalability performance was obtained for the WG-BDDC method on uniform grids by testing over 144 CPUs. Moreover, the WG-BDDC method is successfully extended to fully unstructured grids of all quadrilateral elements to solve beam deformation problems. The results on unstructured grids is comparable qualitatively to that on structured grids. v Table of Contents Acknowledgments .................................. iii Abstract ........................................ iv List of Figures .................................... vii List of Tables ..................................... viii 1 Introduction .................................... 1 2 Numerical Method ................................ 3 2.1 Mathematical Models . 3 2.2 Form Variation . 5 2.3 Weak Galerkin Finite Element Method . 6 2.3.1 Preliminary . 6 2.3.2 Weak Operators . 7 2.3.3 Weak Galerkin Quadrilateral Meshes and Basis Functions . 8 2.3.4 Weak Galerkin Method for Linear Elasticity Equation . 10 3 Parallel Computing Method ........................... 17 3.1 Block Cholesky Elimination and Schur Complement . 17 3.2 FETI Method and FETI-DP Method . 18 3.3 WG-BDDC Method . 22 3.3.1 Primal and Dual Spaces . 23 3.3.2 Preconditioned Conjugate Gradient Method . 25 3.3.3 Local Information Recovery . 26 3.3.4 Local Information Recovery . 27 4 Results and Discussion .............................. 28 4.1 Accuracy Verification . 28 4.1.1 2D Linear Elasticity Equation . 28 4.1.2 Beam Deformation Tests . 31 4.2 Scalability and Speed-up Tests . 34 5 Conclusions and Future Work .......................... 36 A Stabilizer ..................................... 37 B One-dimensional Weak Galerkin Method ................... 39 B.1 Transfer Global Equation into Weak Form . 39 B.2 Weak Gradient Operator . 39 B.3 Stabilizer . 40 B.4 Solution for 1-D Example . 40 Bibliography ..................................... 44 vi List of Figures 2.1 Stress, displacements, and body force . 4 2.2 Weak Galerkin quadrilateral elements and solution points. 8 2.3 Mapping for quadrilateral elements. 9 2.4 Mapping for quadrilateral elements. 15 3.1 Domain decomposition in FETI . 19 3.2 BDDC computational domain . 23 4.1 WG method plotting using structured mesh . 29 4.2 WG method plotting using structured mesh . 30 4.3 Grid deformation after simulation using a structured mesh . 31 4.4 Contour plot the x-component displacements on a structured mesh . 32 4.5 Grid deformation after simulation using an unstructured mesh . 33 4.6 Contour plot the x-component displacements on an unstructured mesh . 33 4.7 Scalability of WGFEM method . 34 B.1 Mesh for 1-D problem . 41 vii List of Tables 4.1 L2 errors for the second order WGFEM on structured grids . 29 4.2 L2 errors for the third order WGFEM on structured grids . 30 4.3 L2 errors for the second order WGFEM on unstructured grids . 31 viii Chapter 1: Introduction Finite Element (FE) methods [2] have been widely used in solving structural mechanics problems over the past 30 years. The FE method can be used on unstructured grids and thus is geometrically flexible. One of the most popular FE methods is the continuous Galerkin finite element method (CGFEM). In CGFEM, the computational domain is divided into a group pf connected elements and unknowns are placed at nodal points of each element. Usually the Galerkin method is used to satisfy the conservation law in a weak form, and the law is satisfied for the global domain rather than for each element. Therefore, CGFEM involves very large, sparse matrices in application. There is a need of accurate parallel computing algorithms for the finite element method because practical engineering problems requires large amount of computational time. One famous parallel computing technique for the standard continuous finite element method is the Finite Element Tearing and Inter Connecting (FETI) [5] accelerated by separating interface unknowns in dual and primal spaces, namely FETI-DP method [3]. In particular numerical analysis, the FETI method is an iterative substructuring method for solving systems of linear equations from the finite element method for the solution of elliptic partial differential equations. In each iteration, FETI [6] requires the solution of a Neumann problem in each substructure and the solution of a coarse problem. This MS thesis research aims to explore future practical applications of the WG method as a new high order discontinuous finite element method. The WG method for solving the second order elliptic equations is developed by Wang and Ye[2], and Mu[3] proved its accuracy on some other partial differential equations such as Biharmonic equations and Helmholtz equations. Usually, quadrilateral elements are favored for solving boundary-layer problems. Therefore, studies in quadrilateral grids instead of triangular grids have been done for the WG method which solves boundary unknowns on element faces. It consists 1 of two key components, i.e., the weak gradient operators contributing stiffness matrix and stabilizer minimizing the differences between interior unknowns and unknowns on the element interfaces through L2 projections [11]. In this research, interior space and boundary space are the same orders of polynomial when making numerical tests, which are the second order and the third order. The simplest version of FETI with no preconditioner in the substructure is scalable with the number of substructures but the condition number grows polynomially with the number of elements per substructure. To avoid this disadvantage, the BDDC algorithm [4], first introduced by Dohrmann, is a variant of the two-level Neumann–Neumann type preconditioner for solving the interface coarse problem. BDDC with a preconditioner consisting of the solution of a Dirichlet problem in each substructure is scalable with the number of substructures and its condition number grows only polylogarithmically with the number of elements per substructure. Throughout this thesis, we will